W9_parallel_resonance.eps

Week 12: Lenses and Mirrors 403

s′=− 6 .7 cm. The magnification ism=−(− 6 .67)/20 = 0.33. The image is erect,

virtual, and smaller than the object. All of these general properties will apply (with

different numbers) toanydiverging mirror.

If you master drawing these generic diagrams (and can manage thevery simple algebra

associated with evaluating e.g.s′andmgivensandf, you can with patience analyze

any combination of mirrors (and later) lenses) you are presented with.

12.4: Lenses

A spherical lensing surface between two different media with different indices of refraction

are drawn in figure (165).

P P’

l

s

r

s’

θ 1

θ 2

n 1 n 2

α β γ

Figure 165: Diagram that shows how a spherical lens creates an image via refraction.

As was the case for the mirror, the three anglesα,β, andγin the small angle approxi-

mation can be written as:

α ≈

ℓ

s

(982)

β = ℓ

r

(983)

γ ≈

ℓ

s′

(984)

We also haveSnell’s lawfor the (small) anglesθ 1 andθ 2 :

n 1 θ 1 ≈n 1 sin(θ 1 ) =n 2 sin(θ 2 )≈n 2 θ 2 (985)

so

θ 2 =

n 1

n 2

θ 1. (986)

Using triangle rules like the ones above, we also get:

θ 1 =α+β (987)

and

β=θ 2 +γ (988)

Eliminatingθ 2 , this becomes:

β=n^1

n 2

θ 1 +γ (989)

If we multiply both sides byn 2 and substituteθ 1 from the first equation, this becomes:

n 2 β=n 1 α+n 1 β+n 2 γ (990)